Questions tagged [mg.metric-geometry]
Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
4,405 questions
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Wasserstein distance in R^d from one dimensional marginals
This question occurred to me while I was reading Klartag's papers on central limit theorems for convex bodies.
Given probability measures $\mu$, $\nu$ on (the Borel $\sigma$-field of) $R^d$ with ...
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Growing a chain of unit-area triangles: Fills the plane?
Define a process to start with a unit-area equilateral triangle,
and at each step glue on another unit-area triangle.
$50$ ...
- 151k
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Needle probing for a convex body
Suppose there is an unknown closed convex body $K$ of
volume vol$(K) = V$ inside the
unit cube $[-\frac{1}{2}, \frac{1}{2}]^d$ in $\mathbb{R}^d$.
You are permitted to probe with a (one-dimensional)
...
- 151k
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Nontrivial trivial integrals
I posted this question to stackexchange and after 24 hours it's got five votes and no answers, so let's see if mathoverflow can say more than that.
Consider two propositions in geometry:
...
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Polyominoes with double contact
Here is a problem which arose from an earlier question. I'll change the terminology but not the question: A polyomino is a region with a connected interior made by joining one or more unit squares ...
- 30.1k
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A topological tree is weakly contractible
Let us call a nonempty topological space a topological tree if it is Hausdorff and for two distinct points there is a continuous injective path connecting the points, which is unique up to ...
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Soft question: mathematics about truchet tiles
It seems that this is the first question on Truchet tiles on MO.
Shown above is a picture of a random tile, which you can see the resulting configuration is much like many membranes of cells.
I ...
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A metric space of geometric shapes
My research involves geometric shapes in $R^2$, and I need a metric with several properties such as:
Families of similar shapes, such as squares, are closed in this metric. Also more general families,...
- 3,549
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Every continuous function is homotopic to a locally Lipschitz one
I would like to know for which category/class/set of metric spaces the following holds: for any two metric spaces $X$, $Y$, for any continuous function $f:X\to Y$ there exists a locally Lipschitz ...
- 1,205
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11
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quantitative version of the rigidity of the 2-sphere
I am looking for a quantitaive version of the following theorem:
A compact surface with $K\equiv 1$ is isometric to the round sphere.
Of course I get the Berger, Brendle-Schoen Theorem which insures ...
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Isoperimetric inequality in complex hyperbolic space
Let $\mathbb{H}_\mathbb{C}^n$ be n-dimensional complex hyperbolic space.
This space is a complex analog of hyperbolic space. It is isometric to the quotient of hyperboloid
$$|z_0|^2-|z_1|^2-\dots-|...
- 1,785
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Is it a coincidence that the universal parabolic constant shows up in the solution to square point picking?
The expected distance $d$ of randomly selected points within a unit square to the square's center is
$d = \frac{1}{6} P$
where P is the universal parabolic constant
$P = \sqrt{2} + \ln{\left(1+\...
- 1,571
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Embeddings of finitely generated groups into uniformly convex Banach spaces
de Cornulier, Tessera, and Valette (Geom. Funct. Anal. 17 (2007), 770-792) conjectured that a finitely generated group $G$ with its word metric admits a bilipschitz embedding into a Hilbert space if ...
- 4,878
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In a locally CAT(k) space, does uniqueness of geodesics imply the lack of conjugate points?
A complete, simply connected Riemannian manifold has no conjugate points if and only if every geodesic is length-minimizing. I just realized that I don't know whether the same is true for a locally ...
- 32.4k
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Sum of squared nearest-neighbor distances between points in a square
Let $\square_2=\{(x,y): 0\leq x, y\leq1\}$ be the unit square in $\mathbb{R}^2$. Take $n>1$ points $P_1, \dots, P_n\in\square_2$.
Denote the distances $d_j=\min\{\Vert P_k-P_j\Vert: k\neq j\}$, ...
- 43.2k
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Tameness in $\mathbb{R}^{n^2}$ of the subset consisting of matrices of positive determinant
The Lie group $GL(n)$ being a manifold is locally path-connected. Consider its connected component of the identity $C\subseteq\mathbb{R}^{n^2}$. What is a good way of showing that $C$ is a tame ...
- 16.6k
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Polygons uniquely inducing arrangements
A beautiful, relatively recent result is that,
Every simple arrangement $\cal{A}$ of $n$ lines in the plane is induced by a simple $n$-gon $P$.
In a simple arrangement, every pair of lines intersect ...
- 151k
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Probability distribution for the number of triangles containing the center of a circle
Pick $n$ points randomly on a circle centered at the origin. Let $X$ be the number of the ${n \choose 3}$ triangles with those vertices that contain the origin in their interior. For fixed $n$, what ...
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How to correctly state Cauchy's rigidity theorem?
Cauchy's rigidity theorem is often stated briefly as
Any two (convex, 3-dimensional) polyhedra with pairwise congruent faces are themselves congruent.
As a more formal generalization to general ...
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Strong equivalence between intrinsic and extrinsic metrics on $GL_n^+$?
$\newcommand{\til}{\tilde}$
Lately, I have become interested in comparing intrinsic and extrinsic metrics on Riemannian manifolds.
Consider $GL_n^+$ (invertible matrices , $\det >0$) as an open ...
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Topological spaces admitting CAT(1) metrics
Suppose that $X$ is a locally contractible completely metrizable topological space. Is it true that $X$ can be metrized as a (complete) CAT(1) metric space?
The only result in this direction I know is ...
- 12.3k
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Is there a version of supersymmetry for homogeneous spaces?
The notion of "supersymmetry" that I am aware of proceeds as follows. One fixes a spacetime $\mathbb R^n$ and signature; I will write $\mathrm{SO}(n)$ for the corresponding group of orthogonal ...
- 54.6k
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Metric conditions on configurations of points with only finitely many solutions
There is an old puzzle, which I believe I learned from Nob Yoshigahara, that asks for all configurations of four (distinct) points in the plane such that the six pairwise distances assume only two ...
- 82.7k
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489
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Shortest morphing between shapes embedded in $\mathbb{R}^3$
I am interested in what in computer graphics is called
morphing between two topologically equivalent shapes $S_0$
and $S_1$ in 3D.
This is a continuous "path" of shapes $S_t$, each embedded and
all ...
- 151k
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702
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Schoenberg's rational polygon problem
"A polygon is said to be rational if all its sides and diagonals are rational, and I. J. Schoenberg has posed the difficult question, ‘Can any given polygon be approximated as closely as we like by a ...
- 151k
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Tiling with incommensurate triangles
Say that two triangles are incommensurate if they do not
share an edge length or a vertex angle, and their areas differ.
Suppose you'd like to tile the plane with pairwise incommensurate triangles.
I ...
- 151k
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Interpret Fourier transform as limit of Fourier series
Let $V=\mathbb{R}^n$, $\Lambda_r=2\pi r \mathbb{Z}^n \subset V (r>0)$ a lattice; $V^*\cong\mathbb{R}^n$ the dual vector space of $V$, and $\Lambda_r^*=\frac{1}{2\pi r} \mathbb{Z}^n =\text{Hom}(\...
- 1,906
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"minimal" embedding of bipartite graphs on a sphere
Here is an easy to pose problem I've encountered (but haven't been able to solve or disprove):
Let (V,E) be a bipartite graph with the following property –
the girth of the graph (i.e. the length of ...
- 1,163
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Are there 100 points that are part of every half-density part of the plane?
Is there a configuration $P$ that consists of 100 points of the plane such that every $X\subset\mathbb R^2$ whose density is half contains an isometric copy of $P$?
I am deliberately being vague ...
- 19k
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John-type theorems: trading structure for accuracy?
Given two symmetric convex bodies $B, B'$ in ${\bf R}^d$, define the Banach-Mazur distance $d(B,B')$ between them to be the least constant $\tau \geq 1$ such that
$$ B \subset TB' \subset \tau B$$
for ...
- 114k
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Can a billiard rack be a square, for every number of balls?
A billiard rack is a rack, usually a triangle, that can hold a certain number of equal size billiard balls, such that the balls' centres cannot move within the rack.
Can the rack be a square, for ...
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Lattices and stable homotopy groups of spheres
The number $65520$ arises in two very different scenarios:
It occurs in the formula for the theta series of the Leech lattice:
$$ \Theta_{\Lambda_{24}}(q) = 1 + \sum\limits_{m=1}^{\infty} \dfrac{...
- 12.4k
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Entropy, magnitude, diversity of finite metric spaces in number theory
I was reading the article by Tom Leinster, (Maximizing
diversity in biology and beyond, arXiv link), and find it very interesting.
Since I was searching for entropies of finite metric spaces I found
...
user6671
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Elkies points in the plane of a triangle $ABC$
Noam Elkies proved that if $x,y,z$ are positive numbers, then there is a unique point $P$ inside $ABC$ such that the inradii $r_a,r_b,r_c$ of the triangles $BPC, CPA, APB,$ respectively, satisfy
$$ ...
- 3,443
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Electrons on a pancake ellipsoid
The problems of minimizing the potential energy of electrons
on a sphere, or maximizing the smallest distance between the electrons,
have been well-studied.
E.g., see the
earlier MO question
"...
- 151k
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Right-angled polytopes
%This question is motivated by the little discussion here at the bottom.
The following thing are known about hyperbolic right-angled polytopes:
Compact hyperbolic right-angled polytopes do not exist ...
- 1,268
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Uniquely geodesic groups
Definition : A group is CAT(0) if it acts properly, cocompactly and isometrically on a CAT(0) space.
Examples : see this blog.
Remark : A CAT(0) space is uniquely geodesic, but the converse is false (...
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When is a submersion locally volume-expanding?
I would like to characterize the smooth maps $\varphi: \mathbb{R}^n \rightarrow \mathbb{R}^k$, $n\geq k$, with the following property:
For every $x\in \mathbb{R}^n$ there exists a positive number $...
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High-dimensional geometry: Top-down Vs. Bottom-up
There are several ways to leverage one's intuition from low-dimensional geometry to understand high-dimensional phenomena. For example, one can get a clearer picture of the behaviour of high-...
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Left invariant metric on ${\rm SL}_n(\mathbb{R})$
I am looking for a left invariant metric on $SL_n(\mathbb{R})$. If this is not possible, it would be acceptable to have a metric on $SL_n(\mathbb{R})/SO_n(\mathbb{R})$ or something like that. Is there ...
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Tiling with similar tiles
Question 1: Is there a polygon $P$ that
cannot tile the plane
and
tiles the plane when copies of $P$ and some other polygon(s) all similar in shape to $P$ but of different size(s) can be used?
...
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Does the centroid depend continuously on the curve?
Let $\gamma$ be a piecewise smooth curve in $\mathbb{R}^n$. Recall that the centroid of $\gamma$ is the point $(\overline{x}, \overline{y})$ where $\overline{x}$ is the average value of $x$ on $\...
- 29.2k
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An introductory text on expanders
I am looking for a book that covers expander graphs rigorously. Preferably a book aimed at beginners.
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Bounding the area of a convex body bounded in a sphere
I have a question which I believe to be pretty basic.
Let $\Gamma$ be some convex body, bounded inside a $L_2$ sphere of radius 1 $B(0,1)$.
Is it true that the surface area of $\Gamma$ is smaller ...
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When does every point in a polytope lie along a chord between its edges?
Consider the 3-simplex, or tetrahedron, in 3-space. Regardless of the positions of the vertices, every point in the simplex lies on a chord between two non-adjacent edges of the simplex. Or, ...
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How can we determine the center of a circle using a straightedge?
Given a circle with diameter AB, how can we determine the center of the circle with a straightedge (we cannot measure lengths, cannot measure angles, or draw parallel lines,... We can only draw ...
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Packing rectangles: Does rotation ever help?
Dominic van der Zypen posed an interesting Box stacking problem.
This is a spin-off question.
Let a collection of rectangles $r_1,\ldots,r_n$ be given by their side lengths in $\mathbb{R}$.
Let $R$ ...
- 151k
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Is this an instance of any existing convex pentagonal tilings?
Inspired by Wikipedia's article on pentagonal tiling, I made my own attempt.
I believe this belongs to the 4-tile lattice category, because it's composed of pentagons pointing towards 4 different ...
- 151
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Dense sphere packings which are not lattice packings
This question is about dense sphere packings in euclidean space $\mathbb R^n$. By a sphere packing I understand any arrangement of mutually disjoint solid open spheres in $\mathbb R^n$, all of the ...
- 4,015